Fractional Knapsack Problem The fractional knapsack problem is a resource allocation problem that deals with assigning items of varying values to a knaps...
The fractional knapsack problem is a resource allocation problem that deals with assigning items of varying values to a knapsack of limited capacity.
Problem Statement:
Given:
A set of items, each with a value and a weight.
A knapsack with a capacity (limited space).
Goal:
Find the fractional allocation (between 0 and 1) for each item that maximizes the total value while staying within the knapsack's capacity.
Example:
Items:
| Item | Value | Weight |
|---|---|---|
| A | 10 | 5 |
| B | 20 | 3 |
| C | 30 | 4 |
Knapsack:
| Capacity |
|---|---|
| 10 |
Optimal Solution:
| Item | Fractional Value |
|---|---|
| A | 0.2 |
| B | 0.3 |
| C | 0.5 |
How it works:
The fractional knapsack problem employs a greedy approach to find an optimal solution. The algorithm works by iterating through the items and adding them to the knapsack until the knapsack is full. For each added item, the algorithm adjusts the fractional values of the remaining items to reflect the changes in total value and capacity.
Key Points:
The fractional knapsack problem is a optimization problem (find the best solution), rather than a search problem (find a solution and check its optimality).
It is a combinatorial problem due to the various choices for allocating items to the knapsack.
The greedy algorithm iterates through items and adds them to the knapsack until it is full, considering fractional values for added items.
The optimal solution for the fractional knapsack is always a fractional allocation (between 0 and 1)